Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

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微分算子环的简单性准则

让ķ是一个具有任意特征的场， $${\cal A}$$ 成为可交换的ķ- 本质上是有限类型域的代数（例如，不可约仿射代数簇上的函数代数）， $${a_r}$$ 成为它的雅可比理想， 和 $${\cal D}\left({\cal A}\right)$$ 是代数上微分算子的代数 $${\cal A}$$ . 本文的目的是为代数给出一个简单的标准 $${\cal D}\left({\cal A}\right)$$ ：代数 $${\cal D}\left({\cal A}\right)$$ 很简单当当 $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A } \right)$$ 对于所有我≥1假设域 K 是完美域. 此外，给出了代数的简单标准 $${\cal D}\left(R\right)$$ 任意交换代数上的微分算子R在任意领域。这给出了一个老问题的答案，即为微分算子的代数找到一个简单标准。